Crystalline condition for $A_{\inf }$-cohomology and ramification bounds

Abstract

For a prime $p>2$ and a smooth proper $p$-adic formal scheme $X$ over ${\mathcal{O}}_K$ where $K$ is a $p$-adic field, we study a series of conditions (${\mathrm{Cr}}_s$), $s\ge 0$ that partially control the $G_K$-action on the image of the associated Breuil-Kisin prismatic cohomology $\mathrm{R}{\Gamma }_{\Delta }(X/\mathfrak{S})$ inside the $A_{\inf }$-prismatic cohomology $\mathrm{R}{\Gamma }_{\Delta }(X_{A_{\inf }}/A_{\inf })$. The condition (${\mathrm{Cr}}_0$) is a criterion for a Breuil-Kisin-Fargues $G_K$-module to induce a crystalline representation used by Gee and Liu, and thus leads to a proof of crystallinity of ${\mathrm{H}}_{\text{eˊt}}^i(X_{\overline{\eta }},{\mathbb{Q}}_p)$ that avoids the crystalline comparison. The higher conditions (${\mathrm{Cr}}_s$) are used to adapt the strategy of Caruso and Liu in order to establish ramification bounds for the mod $p$ representations ${\mathrm{H}}_{\text{eˊt}}^i(X_{\overline{\eta }},\mathbb{Z}/p\mathbb{Z}),$ for arbitrary $e$ and $i$, which extend or improve existing bounds in various situations.

37 pages. Revised version. Added a discussion of finiteness conditions in the construction of Čech-Alexander complexes (Remarks 2.13, 2.17, Lemma 2.18) and appropriate adjustment of related constructions in §4.1. Other minor modifications and updates

References in corpus (2)

• Logarithmic Prismatic Cohomology I
• Comparison of prismatic cohomology and derived de Rham cohomology